Since we are dividing 11 into 40 again, we see that this division will never be exact. We will have 36 repeated as a pattern:

4 11

=

0.363636. . .

By writing three dots (called ellipsis), we mean, "No decimal for

4 11

will ever be complete or exact. However we can approximate it with as many decimal digits as we please according to the indicated pattern; and

the more decimal digits we write, the closer we will be to

4 11

."

That is a fact. It is possisble to witness that decimal approximation and produce it. We have not said that .363636 goes on forever, because we cannot witness that nor can we produce an infinite sequence of digits. It is only an idea.

This writer asserts that what we can actually bring into this world.363636has more being for mathematics than what is only an idea. We can logically produce a decimal approximaiton.

We are taught, of course, that .363636 goes on forever, and so we think that's mathematicsthat's the way things are. We do not realize that it is a human invention.

What is more, infinite decimals are not required to solve any problem in arithmetic or calculus; they have no consequences and therefore they are not even necessary.

Even if we imagine that the decimal did go on forever, then 1) it would never be complete and would never equal ; and 2) it would not be a number. Why not?
Because, like any number, a decimal has a name. It is not that we will never finish naming an infinite sequence of digits. We cannot even begin.

Fractions, then, when expressed as decimals, will be either exact or inexact. Inexact decimals nevertheless exhibit a pattern of digits. The

pattern for

4 11

is .3636363636.

Which fractions -- in lowest terms -- will have exact decimals? Only those whose denominators could be multiplied to become a power of 10. For they are the denominators that a decimal fraction is understood to have. They are the numbers whose only factors are 2's and/or 5's; which are the only factors of the powers of 10.

Here are a few of the numbers that are composed only of 2's or 5's: 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 250, 400. A fraction with any of those denominators will have an exact decimal.

A fraction in lowest terms with denominator 6 will not have an exact decimal, because 6 = 2 × 3. It is not possible to multiply 2 × 3 so that it becomes a power of 10.

Example 4.

a) Show the decimal pattern that

19

generates.

9 goes into 1 zero (0) times.

9 goes into 10 one (1) time with 1 left over.

Again, 9 goes into 10 one (1) time with 1 left over.

And so on. This division will never be exact -- we will keep getting 1's in the quotient.